ag.algebraic geometry - Parametric polynomial solution of a single polynomial equation

Let $P$ be a polynomial in $n$ variables with rational coefficients,
$P in {mathbb Q}[Z_1,Z_2, ldots ,Z_n]$, and consider the algebraic
set
$Z=lbrace (z_1,z_2,z_3, ldots ,z_n) in {mathbb Q}^n |
P(z_1,z_2, ldots ,z_n)=0 rbrace$

If $r$ is a nonnegative
integer, $x_1,x_2, ldots ,x_r$ are variables, and $Q_1,Q_2, ldots ,Q_n$
are polynomials in $x_1,x_2, ldots ,x_r$ such that
$(Q_1(x_1, ldots ,x_r),Q_2(x_1, ldots ,x_r),ldots,Q_n(x_1, ldots ,x_r)) in Z$
for all $(x_1, ldots ,x_r) in {mathbb Q}^r$, we call $(Q_1,Q_2, ldots ,Q_n)$
a $r$-dimensional parametric solution
of the equation $P(z_1,z_2, ldots ,z_n)=0$. It is also
natural to define a maximal parametric solution as one with the largest possible $r$
(to avoid trivialties, we also impose
that there is no variable upon which none of the $Q_i$ depends. I'm not sure
that this last condition avoids all degenerate cases, but I'd like to avoid
definitions that involve advanced notions such as the dimension of an algebraic
variety ).

My questions : is the problem of computing the largest $r$ known to be undecidable in general ? What are the most general cases in which algebraic geometry allows us to compute the largest $r$ (and the associated parametric solutions) ?